Finite semifields with a large nucleus and higher secant varieties to Segre varieties

Michel Lavrauw
2011 Advances in Geometry  
In [2] a geometric construction was given of a finite semifield from a certain configuration of two subspaces with respect to a Desarguesian spread in a finite-dimensional vector space over a finite field. Moreover, it was proved that any finite semifield can be obtained in this way. In [7] we proved that the configuration needed for the geometric construction given in [2] for finite semifields is equivalent with an (n−1)-dimensional subspace skew to a determinantal hypersurface in PG(n 2 − 1,
more » ... ace in PG(n 2 − 1, q), and provided an answer to the isotopism problem in [2] . In this paper we give a generalisation of the BEL-construction using linear sets, and then concentrate on this configuration and the isotopism problem for semifields with nuclei that are larger than its centre. Finite semifields A finite semifield S is a finite division algebra, which is not necessarily associative, i.e., an algebra with at least two elements, and two binary operations + and •, satisfying the following axioms: (S1) (S, +) is a group with identity element 0; A finite field is of course a trivial example of a semifield. The first non-trivial examples of semifields were constructed by Dickon in [5] . One easily shows that the additive group of a semifield is elementary abelian, and the additive order of the elements of S is called the characteristic of S. Contained in a semifield are the following important substructures, all of which are isomorphic to a finite field. The left nucleus N l (S), the middle nucleus *
doi:10.1515/advgeom.2011.014 fatcat:y6pe3vvfbfd65cy5vovpxmm7bq